The Christoffel-minkowski Problem Iii: Existence Problem for Curvature Measures

Curvature measure is one of the basic notion in the theory of convex bodies. Together with surface area measures, they play fundamental roles in the study of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. Let Ω is a bounded convex body in R with C2 boundary M , the corresponding curvature measures and surface area measures of Ω can be defined according to some geometric quantities of M . Let κ = (κ1, · · · , κn) be the principal curvatures of M at point x, let Wk(x) = Sk(κ(x)) be the k-th Weingarten curvature of M at x (where Sk the k-th elementary symmetric function). In particular, W1 is the mean curvature, W2 is the scalar curvature, and Wn is the Gauss-Kronecker curvature. The k-th curvature measure of Ω is defined as Ck(Ω, β) := ∫